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In time-translation-invariant linear response, each infinitesimal past strain step contributes its amplitude times the elapsed-time relaxation modulus. Adding those causal contributions gives the displayed stress-history integral for shear. An equivalent formulation convolves the creep compliance with stress increments. This constitutive law assumes sufficiently small strains about a specified equilibrium or relaxed state; nonlinear deformation, aging and evolving material structure can invalidate that simple superposition.

A four-element linear viscoelasticity model with a spring $E_0$, a Kelvin-Voigt model branch $E_1,{\eta }_1$, and a free dashpot ${\eta }_0$ in series. Adding their strains under common stress gives the displayed creep compliance, with $\tau ={\eta }_1/E_1$. There is an instantaneous elastic response, a delayed recoverable response and steady viscous flow.^[ref] Named after Johannes Martinus Burgers, it extends a Maxwell model with a delayed elastic branch, or a standard linear solid model with a free dashpot.

For a load ${\sigma }_0$ maintained for time $t_h$, the two recoverable contributions behave as in recovery after a finite-duration stress load, while the free dashpot retains ${\sigma }_0{t}_h/{\eta }_0$. HenceThis residual response models viscous flow; it does not by itself establish a plastic yield threshold or a damage mechanism in a real specimen.

A Maxwell fluid has a spring and dashpot in series, so its step-stress creep compliance is $J(t)=1/G_0+t/\eta$. The passive microrheology relation then gives the displayed three-dimensional mean squared displacement. The plateau scale $k_{B}T/(\pi a{G}_0)$ measures elasticity, while the long-time slope $k_{B}T/(\pi a\eta )$ measures viscosity. An entropic network with mesh size $\xi$ has the estimate $G_0\sim k_{B}T/{\xi }^3$, giving plateau scale ${\xi }^3/(\pi a)$.

The nonzero zero-lag extrapolation belongs to the idealized massless spring-dashpot response. Actual displacement variance starts at zero; fast solvent dynamics or inertia regularize the short-time behavior. Consequently the plateau formula applies after those transients and before the Maxwell relaxation time, if those scales are well separated. A sampling interval comparable with $\tau$ gives little access to a genuine plateau, even when spatial localization is sufficiently precise.

For a dilute spherical no-slip probe of radius $a$ in an isotropic incompressible linear medium at equilibrium, the bead's displacement per unit step force is $J_b(t)=J(t)/(6\pi a)$, where $J$ is the bulk shear creep compliance. The fluctuation-dissipation theorem gives one-coordinate displacement variance $2k_{B}T{J}_b(t)$. Adding the three equal Cartesian contributions proves the displayed relation. In a Newtonian fluid, $J=t/\eta$, so it becomes $⟨\Delta r^2⟩=6Dt$ with the Stokes–Einstein relation $D=k_{B}T/(6\pi \eta a)$.

The particle must sample a continuum response on its scale and remain in thermal equilibrium. A two-dimensional image records only two of the three displacement components; isotropy then gives $⟨\Delta r^2{⟩}_{3d}=(3/2)⟨\Delta r^2{⟩}_{2d}$. Localization noise, confinement and unresolved fast relaxation can otherwise be mistaken for elastic response.

A simple fluid reduces to a linear viscoelastic fluid when its deformation and rotation over the relevant memory interval are small enough to linearize the constitutive history about an isotropic relaxed state. Small amplitude is essential; small frequency is not. For example, arbitrarily high-frequency oscillation can still be linear if its strain amplitude is sufficiently small. The relaxation modulus $G(t)$ is the shear-stress response to a unit small step of shear strain.

Choose the harmonic convention $e^{i\omega t}$. If the imposed shear rate is $\mathrm{Re}(\hat{\dot{\gamma }}{e}^{i\omega t})$, the corresponding shear stress is $\mathrm{Re}({\eta }^{\ast }(\omega )\hat{\dot{\gamma }}{e}^{i\omega t})$. Substitution into the constitutive convolution gives the complex viscosity:For a nondecaying elastic part this formula is understood through the limit from a positive Laplace damping parameter. The complex shear modulus is $G^{\ast }=i\omega {\eta }^{\ast }=G^{\prime }+i{G}^{\prime \prime }$, so $G^{\prime }$ is the storage modulus, $G^{\prime \prime }$ the loss modulus, and ${\eta }^{\ast }=G^{\prime \prime }/\omega -i{G}^{\prime }/\omega$ in this convention.

Creep is the progressive deformation produced by a held stress. For a small step $S_{0}H(t)$ of shear stress, define the creep compliance $J$ by $\gamma (t)=S_{0}J(t)$ from a relaxed state. Writing hats for Laplace transforms, the same constitutive law gives $\hat{S}=p\hat{G}\hat{\gamma }$. Since $\hat{S}=S_0/p$, this yieldsHere $H$ is the Heaviside step function. Inversion determines the full creep response, including any instantaneous elastic strain jump. Equivalently, the causal relaxation/creep convolution includes that jump through ${\int }_{0^{-}}^{t}G(t-s)dJ(s)=1$ for $t>0$.

For the Maxwell fluid, $\hat{G}(p)=G_0/(p+1/\tau )$. Thus the two material responses areThe creep response is an elastic jump followed by constant-rate viscous deformation. The zero-frequency shear viscosity is $\eta =G_0\tau$.

For the wall problem, take $v=u(y,t)e_x$ and denote the shear component of stress by $S$. The incompressibility constraint is satisfied, and the advective part of the material derivative vanishes because no field depends on $x$. With no imposed longitudinal pressure gradient, momentum conservation and the linear Maxwell law giveThe initial conditions are $u=S=0$ in $y>0$; the no-slip boundary condition is $u(0,t)=UH(t)$, with decay as $y\to \infty$. Transforming these equations givesChoose the analytic square-root branch in $\mathrm{Re}p>0$ with $\mathrm{Re}q>0$, so that the spatially increasing solution is excluded. The Maxwell start-up shear layer consequently hasThe Bromwich inversion formula now givesThe vertical Bromwich contour is directed upward and lies to the right of all singularities, including the origin and the branch points at $0$ and $-1/\tau$. A cut joining these branch points is a possible continuation into the left half-plane; inversion only requires the stated right-half-plane branch.

The Newtonian limit holds $\eta =G_0\tau$ fixed while $\tau \to 0$, so $G_0\to \infty$. Then $q(p)\to \sqrt{\rho p/\eta }$. With $\nu =\eta /\rho$, the inverse Laplace transform gives the complementary error function profileThis is diffusive momentum penetration in a Newtonian fluid, with thickness of order $\sqrt{\nu t}$.

The elastic limit instead holds $G_0$ fixed while $\tau \to \infty$. Write $c=\sqrt{G_0/\rho }$. On the chosen branch, $q(p)\to p/c$, so the Laplace transform time-shift rule yieldsAn elastic shear front travels at speed $c$: the material ahead is at rest, while that behind moves at the wall speed. Indeed the limiting displacement is $d(y,t)=U(t-y/c)H(t-y/c)$, whose shear strain is $d_y=-(U/c)H(t-y/c)$. A nonzero pointwise velocity gradient at the front is interpreted distributionally, or by smoothing the ideal wall start.

The use of linear viscoelasticity is conditional, rather than automatically justified for arbitrary $U$. The material derivative simplifying to a time derivative does not remove the nonlinear constitutive effects of finite shear strain. In this Maxwell problem the rapid elastic response supplies the natural strain scale $U/c=U\sqrt{\rho /G_0}$, which must be small for linearization through start-up. Away from the front, the strain accumulated over an effective memory interval must likewise be small. In the late diffusive regime its characteristic size is $\tau U/\sqrt{\nu t}$, which decreases with time. Large absolute wall displacement $Ut$ alone is not a disqualification for a fluid with finite memory; it is relative deformation during that memory that matters. In the elastic limit, where memory does not decay, the explicit strain behind the front confirms the condition $U/c\ll 1$. Without such a small-strain restriction, the computed field is a formal linear response and need not describe an arbitrary simple fluid.

In a material creep experiment, apply a small step shear stress ${\sigma }_{0}H(t)$ and measure the resulting strain. The creep compliance is $J(t)=\gamma (t)/{\sigma }_0$. An elastic solid reaches a bounded strain; a fluid can continue deforming.

In a stress-relaxation experiment, impose a small strain step ${\gamma }_{0}H(t)$ and measure stress while holding the deformation. The relaxation modulus is $G_{\mathrm{rel}}(t)=\sigma (t)/{\gamma }_0$ after the loading transient. A fluid's stored shear stress can decay to zero, while an elastic solid retains a nonzero long-time contribution.

The original curves distinguish the controlled input from measured output. The strain/stress must be small enough for linear response, and real finite loading rise time, instrument compliance, wall slip and temperature must be characterized. Otherwise an instrumental transient can be mistaken for a material relaxation mode.

The creep compliance is the strain per unit step stress: if $\sigma (t)={\sigma }_{0}H(t)$, then $\gamma (t)={\sigma }_{0}J(t)$ for $t>0$. A Newtonian fluid obeys $\sigma =\eta \dot{\gamma }$, so under constant stress its initially zero strain grows as $\gamma ={\sigma }_{0}t/\eta$. Therefore

A creep experiment applies a small step of stress and holds it constant while recording strain. For $\sigma ={\sigma }_{0}H(t)$, the series spring jumps by ${\sigma }_0/G_0$ and the dashpot subsequently flows at constant rate ${\sigma }_0/\eta$. The creep compliance is thusUsing the stipulated effective stress and strain substitutions givesOn linear axes the model is a straight line of positive intercept $A=k_{B}T/(\pi a{G}_0)$ and slope $A/\tau$. On logarithmic axes it has an approximate elastic plateau for $t\ll \tau$ and slope one for $t\gg \tau$, within the time window where the continuum approximation applies. The long-time slope is $6D$ with $D=k_{B}T/(6\pi \eta a)$, agreeing with the Stokes–Einstein relation.

There is an essential zero-lag convention. A literal lag displacement $\Delta \mathbf{r}(t)=\mathbf{r}(t_0+t)-\mathbf{r}(t_0)$ has $⟨|\Delta \mathbf{r}(0){|}^2⟩=0$. The nonzero intercept above is the spring-compliance extrapolation of the idealized microrheology of a Maxwell fluid, after unresolved solvent or inertial transients. It must not be interpreted as an instantaneous finite physical displacement at exactly zero lag. The stated mesh condition motivates a continuum probe response; quantitative use is more reliable for $a\gg m$ than merely $a>m$.

A viscoelastic material stores part of the deformation energy and dissipates part of it. In linear viscoelasticity, a small imposed shear strain produces a stress determined by its history, $\tau (t)={\int }_{-\infty }^{t}G(t-t^{\prime })\dot{\gamma }(t^{\prime })d{t}^{\prime }$. A shear step probes stress relaxation; a stress step probes creep compliance. A purely elastic solid has persistent stress after a held strain, while a Newtonian fluid relaxes that stress and creeps continuously under constant stress.

For a Maxwell model, a spring of shear modulus $G_0$ and a dashpot of viscosity $\eta$ are in series:For the convention $\gamma \propto e^{i\omega t}$,The storage modulus dominates at short times or high frequencies; the loss modulus dominates at long times or low frequencies. The energy dissipated per cycle per volume is $\pi G^{\prime \prime }{\gamma }_0^2$. A Kelvin-Voigt model places spring and dashpot in parallel and instead gives bounded creep $J(t)=(1-e^{-G_{0}t/\eta })/G_0$: a simple illustration of a viscoelastic solid. Real materials can require a spectrum of relaxation times rather than one element.

A small-amplitude oscillatory rheometer measures stress amplitude and phase relative to imposed strain, giving $G^{\prime }=|{\tau }_0/{\gamma }_0|\cos \delta$ and $G^{\prime \prime }=|{\tau }_0/{\gamma }_0|\sin \delta$. First sweep strain amplitude to identify the linear range; control temperature, sample ageing, wall slip and instrument inertia. Frequency-dependent phase lag alone is insufficient if these apparatus effects are unresolved.

Use a rheometer, for example a cone-and-plate rheometer or a parallel-plate rheometer, to impose a small deformation and measure the required torque, or impose a torque and measure the deformation. The apparatus geometry converts these into shear stress and strain. First check that reducing the imposed amplitude does not change the normalized response; this identifies the linear regime and avoids yielding or structural rearrangement.

Three complementary protocols characterize the material's memory. In a stress relaxation experiment impose a small step ${ϵ}_{0}H(t)$ and hold the deformation fixed. The output is the relaxation modulus $G(t)=\sigma (t)/{ϵ}_0$, revealing how stored stress decays and whether an elastic plateau remains. In a creep compliance measurement impose a small stress step ${\sigma }_{0}H(t)$ and record $J(t)=ϵ(t)/{\sigma }_0$. Removing the load also distinguishes recoverable deformation from permanent flow.

In small-amplitude oscillatory shear, impose sinusoidal strain over a range of frequencies and measure both stress amplitude and phase. With $ϵ={ϵ}_0\cos \omega t$, fitThe outputs are the storage modulus $G^{\prime }$ and loss modulus $G^{\prime \prime }$, or equivalently amplitude ratio and loss tangent $\tan \delta =G^{\prime \prime }/G^{\prime }$. These measure elastic storage and dissipative response, respectively. The step experiments give time-domain information and the oscillatory experiment gives the corresponding frequency-domain response. All comparisons must use the same deformation convention and controlled temperature.

A Kelvin-Voigt model has no finite instantaneous elastic compliance: the viscous element prevents a sudden deformation under a finite stress step. Its creep compliance isConversely an ideal strain step requires an impulsive stress,After the impulse the held-strain response is simply constant. There is no finite stress-decay interval, and the loss modulus grows without bound as $\omega \to \infty$. This fails to represent a material with both a finite instantaneous stiffness and delayed stress relaxation.

The added series spring resolves this particular defect. At the instant of loading the Kelvin element cannot move but the added spring can. The standard linear solid model consequently hasandIts instantaneous stiffness is $G_H=G_1$, its long-time stiffness is $G_L$, and it exhibits genuine stress relaxation as well as delayed creep. The creep retardation time ${\eta }_0/G_0$ differs from the stress-relaxation time $\tau ={\eta }_0/(G_0+G_1)$. These properties also follow directly from its constitutive equation,It remains an idealized solid with one relaxation mode: it cannot describe unrestricted long-time viscous flow, nonlinear response, or a broad spectrum of relaxation times.

A spring $E_0$ in series with a parallel spring-dashpot branch $E_1,\eta$ has creep compliance $J(t)=1/E_0+(1/E_1)(1-e^{-E_{1}t/\eta })$. It includes immediate and delayed recoverable strain, but no unbounded viscous flow. Large strain, damage or anisotropy require additional constitutive structure.

A standard linear solid model has a finite instantaneous stiffness, a nonzero long-time stiffness and one exponential stress-relaxation mode. A Kelvin-Voigt model with spring $G_0$ and viscosity $\eta$ in series with spring $G_1$ givesThese expressions follow by adding series strains under common stress. Its storage modulus is $G_L+(G_H-G_L)x^2/(1+x^2)$ and its loss modulus is $(G_H-G_L)x/(1+x^2)$ for $x=\omega \tau$. Its relaxation modulus is $G(t)=G_L+(G_H-G_L)e^{-t/\tau }$, while its creep compliance is $J(t)=G_1^{-1}+G_0^{-1}(1-e^{-G_{0}t/\eta })$. Thus creep retardation and stress-relaxation times differ. This solid has bounded creep and cannot represent terminal fluid flow.